Directed Ramsey number for trees

Abstract

In this paper, we study Ramsey-type problems for directed graphs. We first consider the k-colour oriented Ramsey number of H, denoted by − → r (H, k), which is the least n for which every k-edgecoloured tournament on n vertices contains a monochromatic copy of H. We prove that − → r (T, k) ≤ c k |T | k for any oriented tree T . This is a generalisation of a similar result for directed paths by Chvátal and by Gyárfás and Lehel, and answers a question of Yuster. In general, it is tight up to a constant factor.We also consider the k-colour directed Ramsey number ← → r (H, k) of H, which is defined as above, but, instead of colouring tournaments, we colour the complete directed graph of order n. Here we show that ← → r (T, k) ≤ c k |T | k−1 for any oriented tree T , which is again tight up to a constant factor, and it generalises a result by Williamson and by Gyárfás and Lehel who determined the 2-colour directed Ramsey number of directed paths.

Monochromatic trees in random tournaments

“…We will make use of the following theorem (Theorem 3.17 in [4]). In fact, we will only use a special case of this theorem, when one of the trees is a path.…”

“…10. Let 0 < δ < 1 4 . In every oriented graph G with at least δ|G| 2 edges, there is a δ 4 |G|-mindegree pair.…”

“…With this in mind, it is natural to ask if the tight bound for a tree T depends only on the order of the tree and the length of its longest directed subpath, denoted by (T ). More precisely, Bucić, Letzter and Sudakov [4] ask if the directed Ramsey number of a tree is O(|T | · (T )); if this holds, it can readily be seen to be tight. They prove that this holds for oriented paths.…”

“…It is interesting to consider oriented Ramsey numbers of further acyclic graphs, and the natural next example are trees. It turns out that oriented trees behave similarly to paths in terms of their oriented Ramsey numbers: it was proved by Bucić, Letzter and Sudakov [4] that given any (oriented) tree T on n vertices and any tournament G on cn 2 vertices (where c is a positive constant), we have G → T , i.e. the oriented Ramsey number of any tree of order n is at most cn 2 .…”

Monochromatic paths in random tournaments

“…In a forthcoming paper , we further generalize Theorem 1 to oriented trees: we show that given any oriented tree T on n vertices, with high probability in any 2‐coloring of a random tournament T on vertices, we can find a monochromatic copy of T . To achieve this we use many ideas and concepts of this paper combined with ideas from , as well as some new ideas. This bound is tight up to a polylog factor ( n – 1 is a trivial lower bound).…”